According to Euripides, the dowry of Klytaimnestra [Clytaemnestra/ Clytemnestra]1 is a male slave and some other undisclosed items (and/or persons?).
We do not have any such detail(s) about what Helen's dowry is, unless we guess that the entire kingdom of Lakedaimon is this dowry. I'm not convinced that such is necessarily the case. (More on that below.)
From his father Atreus, Agamemnon inherits the kingdom of Argolis, which is already wealthy and powerful at the time of his accession to the throne thereof. He goes on to augment its power through his own agency.
There is no obvious connection between his marriage and his status, except insofar as it appears to be strategically beneficial in terms of political alliance. On top of that, things are yet easier for him in this regard because his brother Menelaus, with whom he is on good terms, has already married into the same family by the time that Agamemnon weds Klytaimnestra.
Apart from gaining a high-class wife, as well as further consolidation of power between Argolis and Lakedaimon, somehow (it seems counter-intuitive, at least from a regional diplomacy point-of-view, to start it off by widowing the prospective bride and killing her baby), I don't see that Agamemnon has anything to gain by marrying Klytaimnestra.
Based on all this, his status does not seem to depend on his wife, royal or otherwise. Granted, of course, that upon feeling slighted enough by him, this wife does organise for him to be slaughtered when he returns home from Troy, so in that sense, well... his status depends on her a frightfully good deal.
At the end of the 400s BC Euripides wrote a play entitled Iphigeneia at Aulis, in which a fairly major character is a quirky unnamed individual known simply as the "Old Man."
In the play's introduction scene, it is pointed out that the Old Man is a slave belonging to the royal household of Argolis. Via a bit of expository dialogue in Lines 42-45—addressed here to Agamemnon but really directed at the audience—the slave reveals to us that:
Your old father-in-law Tyndareos [Tyndareus] gave me to your wife Klytaimnestra
as part of her dowry, to be her loyal servant.
The Old Man makes a number of subsequent appearances on stage. In his final scene he reminds us—this time addressing Klytaimnestra—that, "as you know, I was part of your dowry when you married King Agamemnon" (Line 869).
These are the only explicit ancient references I have found to the content of Klytaimnestra's dowry.2 The elderly slave makes no mention of what (or who) else accompanied him at the time of the transfer in question.
It is arguably a lot messier to determine what Helen's dowry was. It could perhaps be said that her dowry was the entire kingdom of Lakedaimon, into which Helen had been born a princess, but I don't see any explicit mention which draws that conclusion.
It is a (maybe surprisingly) common trope in Greek mythology for a foreign man to marry into a city's royal family and thus become king, succeeding his father-in-law on the throne. Typically this is because the father-in-law has produced no direct male heirs or he has survived all of his male issue.
(See for example Perseus inheriting the throne of Ethiopia, which he never takes but transfers to his firstborn son Perses; Pelops—grandfather of Agamemnon and Menelaus—becoming king of Pisa after Oinomaos [Oenomaus]3; Alkathous [Alcathous], a son of Pelops [and thus uncle of Agamemnon and Menelaus] becoming king of Megara after Megareus; Sikyon [Sicyon], another son of Pelops, succeeding Lamedon as king of Mekone, which he renames Sikyon after himself; and [Priam's ancestor] Dardanos [Dardanus] becoming king of Teukria [Teucria] after Teukros [Teucer].)
Are we to automatically understand from this that the kingdom which the outsider inherits is ordinarily part of his princess bride's dowry? If the answer to that is no, then we do not seem to be told—between the entire kingdom of Lakedaimon on the one hand, and merely a fraction of its resources on the other—where Helen's dowry begins and ends.
The Status of Menelaus
But let's assume that the answer to that is yes, and that the dowry in question includes the lion's share of Tyndareos' family wealth. How Helen and her own position should affect the status of Menelaus is still a murky thing to ascertain, not least because of the unprecedented saga in which the two of them get embroiled.
It obviously is an unusual nuptial negotiation setup to have all of Greece's most powerful and most important kings and princes suing for the hand of one princess at the same time with as much determination as the suits for Helen end up garnering.
Nonetheless, even with all of that baggage to deal with, there is a scene in Iphigeneia at Aulis which suggests that Menelaus bears no obligation to recover his wife from Paris. At his first appearance in the play, Menelaus engages Agamemnon in a quarrel about the planned sacrifice of Agamemnon's daughter Iphigeneia to Artemis in order for the goddess to release the winds for the Greek fleet to sail to Troy and attack it.
Agamemnon has been trying to find a way to avoid killing his daughter, and Menelaus berates him for it. They are interrupted by a messenger's announcement of the arrival of Iphigeneia, who believes that she's been summoned in order to wed the Myrmidon leader Achilles.
Upon Agamemnon's evident anguish that his daughter is now in Aulis, Menelaus changes his mind and even urges Agamemnon not to go through with it. They could just disband the gathered army and forget the whole expedition. As he says:
After all, what is it exactly that I want? Is it marriage? Surely I
could find another wife elsewhere! Gods forbid that I should choose to
lose a brother to win a Helen! Should I exchange the good for bad?
He doesn't seem much concerned about losing the throne, if such a threat even exists. Another wrinkle to the question of Lakedaimon's throne has already appeared. Menelaus and Helen have a daughter, named Hermione, whom Helen left behind in Lakedaimon when she took off with Paris.
Is Hermione a secure heir at this point, whether her mother is a present and dutiful wife or if her mother has been taken by another man? Is it enough that Menelaus has had a legitimate daughter by the previous heiress to the kingdom, whatever it is that may have happened to said heiress?
Going by what seems to be virtually customary all over Greece and the Mediterranean Basin by this point, as far as the Greek myths go, all we would need to do is find an eligible (typically foreign and royal) bachelor for Hermione to wed and it's all good: he would be the next king. And that need not mean Menelaus' displacement before his time is done.
As fate would have it, later on, after the saga has cooled down and Helen has been brought back, a stranger (in a sense) comes to town and marries Hermione. The groom is none other than Prince Orestes of Argolis, the son of Agamemnon and Klytaimnestra, and thus Hermione's own cousin, who will thus rule over both Argolis and Lakedaimon in a consolidated position now stronger than either of the immediately preceding rulers of these kingdoms.
Going back to Menelaus: perhaps the lines are blurry enough between wounded pride; the famous Oath of Tyndareos; and doubt as to whether his position on the throne is secure after he has been so vilely cuckolded and robbed of "the treasures" (which are usually invoked as part of what the Greeks demand to be returned along with Helen) by an honoured guest in his house, that, by the conclusion, his brother Agamemnon is convinced of his obligation to sacrifice Iphigeneia. Thereafter, of course, come the long years of war far from home.
The Status of Agamemnon
While Menelaus' status might be quite open to interpretation, I think that Agamemnon's is much less so. Apart from the apparent politically diplomatic (and thereby strategic?) connection to his wife's family and its fame, Agamemnon does not appear to have anything to gain by marrying Klytaimnestra.
Agamemnon is primarily the king of the city of Mykenai [Mycenae], which position makes him the ruler of all of Argolis, a large northeastern swath of Peloponnesos [Peloponnesus] which contains the great cities of Argos, Tiryns and Troizenos [Troezen]. The rule of the city of Argos is divided among at least four kings (including Kyanippos son of Adrastos; Sthenelos son of Iphis; and Tlepolemos son of Herakles) who reign under Agamemnon. By the time the Trojan War has ended (or shortly after) so have the reigns of these men, unceremoniously, while the throne of Agamemnon is still very much intact, to later devolve upon his son Orestes, who becomes even more powerful than his father.4
Shortly after becoming king, Agamemnon led an army north of Argos, against King Hippolytos [Hippolytus] of Sikyon, a great-grandson of Herakles, and made him subject to Mykenai. According to Strabo (Geographika 8):
When everything fell to the sons of Atreus, Agamemnon, being the
elder, assumed the supreme power, and by a combination of good fortune
and valour acquired much of the country in addition to the possessions
he already had; and indeed he also added Lakonia to the territory of
Mykenai. Now Menelaus came into possession of Lakonia, but Agamemnon
received Mykenai and the regions as far as Korinthos [Corinth] and
Sikyon and the country which at that time was called the land of the
Ionians and Aegialians but later the land of the Achaeans.
Lakedaimon was part of Lakonia. Strabo's account here essentially nullifies altogether the idea that Menelaus was dependent on his father-in-law for his position as king of Lakedaimon. Apollodorus' Epitome 2.15, however, tells a somewhat different story in which Tyndareos, at least for a time, looked after Agamemnon and Menelaus when they were in exile from Argolis, saying that he brought them back from their wanderings in Sikyon and Aitolia [Aetolia] so that they could reclaim their birthright. Perhaps there was some symbiosis taking place in their relationship with Tyndareos.
Beyond all that, Agamemnon held territory near Pylos in Messenia (on the direct opposite side of Peloponnesos from his home base), at least seven cities' worth, and, according to Strabo, he imposed his rule over all of Peloponnesos, like his grandfather Pelops, who had (re)named the vast southern peninsula of Greece, "Pelops' Island," after himself, its emperor.
It is remarkable that Agamemnon gets away with the murder of Klytaimnestra's former husband, and the manner in which he manages to do so is noteworthy. In Iphigeneia at Aulis 1150-1155, Klytaimnestra's divine brothers the twin Dioskouroi [Dioscuri], who had departed from the world of mortals years before, are so enraged at Agamemnon's slaying of their brother-in-law and their nephew that they actually leave Zeus' side (presumably in heaven or on Mt Olympos [Olympus]) and come charging on immortal horseback against Agamemnon.
Inexplicably, Agamemnon pleads with Tyndareos to help him out, and Tyndareos obliges him, not only convincing the Dioskouroi to forgive the offence but also marrying Klytaimnestra off to her husband's killer! Either Agamemnon had something on Tyndareos or they must have had a stellar relationship.
None of these events, however, seem to add to Agamemnon's clout beyond the ways already aforementioned. He was already a rich and powerful king and his dominions continued to increase after he got married, evidently in spite of the marriage. As pointed out previously, his brother already ruled the southeastern portion of Peloponnesos (which Strabo understands Agamemnon to have given to him in the first place anyway), which certainly would only have served him further.
Again, not that any of this saved him from a bedchamber assassination plot when he got back home from the war in Asia.
The Murder of Tantalos [Tantalus]
I can think of one theoretical motive behind Agamemnon's actions surrounding his marriage to Klytaimnestra. As alluded to above, Agamemnon acquired Klytaimnestra as a bride by killing the man to whom she was already wed at the time that Agamemnon forcibly betrothed her to himself.
This man was named Tantalos, whom Pausanias tells us was the son of either Thyestes or Broteas. Thyestes was the brother of Atreus, thus making the former rendition of Tantalos a cousin of Agamemnon. Broteas was the brother of Pelops, so in the latter case Tantalos would have been from the previous generation, as a cousin of Atreus.
Wikipedia claims that this Tantalos was king either of Lydia in Asia Minor or of the city of Pisa on Peloponnesos. It provides no reference(s) for the claim regarding Lydia but as far as Pisa is concerned, Iphigeneia at Aulis is cited as the source. The only two mentions of Tantalos in this play, however, are: a cursory reference to the first Tantalos—Pelops' and Broteas' father, after whom this second one (his grandson or great-grandson) is named; and when Klytaimnestra reminds Agamemnon that he had killed her first husband who bore this name. Nothing is said about anyone ruling Lydia, and Pisa appears nowhere in the script.
Wikipedia therefore seems to be confounding the first Tantalos with the second in calling him king of Lydia. As for him being king of Pisa, it is not an unreasonable conclusion, since, if Thyestes is Tantalos' father, then Thyestes' father Pelops would have preceded Tantalos on that city's throne. Nevertheless, Wikipedia might be first place at which such a conclusion has been arrived.
It would have been quite helpful if we did have some ancient document corroborating these claims, whereupon we could suggest that perhaps Agamemnon killed Tantalos as an act of royal usurpation. Even this would fall short of much substance, however, since Agamemnon never comes to have any interest whatsoever in the faraway Lydia. Pisa, on the other hand, never comes up as carrying especial importance, even as it falls under the purview of Agamemnon when he becomes dominant over the whole of Pelops' Island.
A more compelling version of our Tantalos here could be if he were the son of Thyestes and perceived by Agamemnon as a potential threat to his hold on the throne of Argolis. A jumble of other traditions recount the nasty rivalry between Atreus and Thyestes before the births of their children, culminating in the death of Thyestes, and the ascendancy of Atreus and his offspring. That rivalry had been centred mostly on a fight for sovereignty over Argos. It makes a lot of sense for there to be bad blood between the heirs of Atreus and Thyestes, here represented by Agamemnon and Tantalos.
This would have sealed the deal for your Question, I think, if we knew that Tantalos had been king of Argos (or even of some other minor city in Argolis) and that Agamemnon killed him specifically for this reason. From there it would take a short step to assert that Agamemnon may have needed this king's widow to cement himself into position as the new ruler.
But none of this is the case. For the slaying of Tantalos, we are never supplied with a motive on the part of Agamemnon, who we know to already have been the unrivalled heir to the throne, if not already king, of Argolis. The preceding guesses make for a more fleshed out back story, but they are merely that: my guesswork.
As far as the marriage aspect of this is concerned, it may be simply that Agamemnon was so enamoured of Klytaimnestra that he determined to procure her regardless of the collateral damage. But neither is this spelled out so starkly or clearly anywhere. Aside from the (modest?) dowry and potential minor fringe benefits aforementioned, and the more mundane reasons that anyone else would have for engaging in nuptials, Agamemnon gets nothing from his bride he doesn't already have.
1. Okay, so technically the true ancient spelling of this character's name, such as it occurs, for instance in Iphigeneia at Aulis, actually has no N in it. It is really Κλυταιμήστρα = Klytaimestra/ Clyt[a]emestra. The more common spelling in modern times (with the N), such as I have opted to use in this Answer, supposedly does not occur before the 700s AD, at the earliest.
2. It would be interesting if there are after all other references which provide further detail.
3. In the case of Pelops and Oinomaos, the only way for Pelops to marry Oinomaos' daughter is by, first, defeating his father-in-law-to-be in a chariot-race and then killing Oinomaos, which tasks Pelops succeeds in performing (even if he cheats a little bit in order to win).
4. Between Agamemnon's death and Orestes' accession to the throne, though, there is a sort of interregnum in which Kylarabes son of Sthenelos rules Argolis.